# Factors, Multiple and Primes - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Factors, Multiples and Primes.

Printable worksheets containing selections of these problems are available here:

 Stage 3 ★ Sheet 1 Solutions Stage 3 ★★★ Sheet 1 Solutions Sheet 2 Solutions Sheet 3 Solutions Stage 3 ★★ Sheet 1 Solutions Stage 4 ★★★ Sheet 1 Solutions Sheet 2 Solutions Sheet 3 Solutions Sheet 4 Solutions

### Last-but-one

##### KS 3 Short Challenge Level:

Weekly Problem 1 - 2016
What is the last-but-one digit of 99! ?

### What's on the Back?

##### KS 3 Short Challenge Level:

Weekly Problem 21 - 2017
Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?

### Almost a Million

##### KS 3 Short Challenge Level:

Weekly Problem 26 - 2014
Which of the given numbers is divisible by 6?

### Multiplication Table Puzzle

##### KS 3 Short Challenge Level:

Weekly Problem 6 - 2017
In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?

##### KS 3 Short Challenge Level:

Weekly Problem 5 - 2011
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

### Grandma's Cake

##### KS 3 Short Challenge Level:

Weekly Problem 7 - 2016
What is the smallest number of pieces grandma should cut her cake into to guarentee each grandchild gets the same amount of cake and none is left over.

### Calculation 2000

##### KS 3 Short Challenge Level:

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

### Peter's Primes

##### KS 3 Short Challenge Level:

Weekly Problem 22 - 2017
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

### Multiple Years

##### KS 3 Short Challenge Level:

Weekly Problem 18 - 2016
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?

### Prime Order

##### KS 3 Short Challenge Level:

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

### Reversible Primes

##### KS 3 Short Challenge Level:

How many two-digit primes are there between 10 and 99 which are also prime when reversed?

### End of a Prime

##### KS 3 Short Challenge Level:

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

### Essential Supplies

##### KS 3 Short Challenge Level:

Weekly Problem 25 - 2006
Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

### Producing Zeros

##### KS 3 Short Challenge Level:

Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

### Stamp Collecting

##### KS 3 Short Challenge Level:

Last week, Tom and Sophie bought some stamps and they spent exactly £10. Can you work out how many stamps they bought?

### Tricky Customer

##### KS 3 Short Challenge Level:

Weekly Problem 6 - 2015
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

### Flora the Florist

##### KS 3 Short Challenge Level:

Weekly Problem 45 - 2009
Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

### Punky Fish

##### KS 3 Short Challenge Level:

Weekly Problem 12 - 2008
A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?

### Threes and Fours

##### KS 3 Short Challenge Level:

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

### Product 100

##### KS 3 Short Challenge Level:

Weekly Problem 38 - 2010
The product of four different positive integers is 100. What is the sum of these four integers?

### Relative Time

##### KS 3 Short Challenge Level:

Weekly Problem 25 - 2014
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

### Divisible Digits

##### KS 3 Short Challenge Level:

Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?

### Pairing Up

##### KS 3 Short Challenge Level:

Weekly Problem 14 - 2008
The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

### Eight Factors Only

##### KS 3 Short Challenge Level:

We are given two factors of a number with eight factors. Can you work out the other factors of the number?

### Find from Factors

##### KS 3 Short Challenge Level:

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

### Back of the Queue

##### KS 3 Short Challenge Level:

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

### Red Card Blue Card

##### KS 3 Short Challenge Level:

Can you arrange the red and blue cards so that the rules are all followed?

### Common Remainder

##### KS 3 Short Challenge Level:

144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?

### School Dinners

##### KS 3 Short Challenge Level:

Weekly Problem 1 - 2009
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?

### Sticky Fingers

##### KS 3 Short Challenge Level:

Weekly Problem 7 - 2011
Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?

### Coin Collection

##### KS 3 Short Challenge Level:

Weekly Problem 5 - 2007
When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?

### One Short

##### KS 3 Short Challenge Level:

Weekly Problem 39 - 2014
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

### Trailing Zeros

##### KS 3 Short Challenge Level:

Weekly Problem 13 - 2009
How many zeros does 50! have at the end?

### Square Sum

##### KS 3 Short Challenge Level:

Weekly Problem 32 - 2007
One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

### Indivisible

##### KS 3 Short Challenge Level:

Weekly Problem 25 - 2011
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

### Powerful Zeros

##### KS 3 Short Challenge Level:

Weekly Problem 32 - 2006
How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?

### Smallest Abundant Number

##### KS 3 Short Challenge Level:

Weekly Problem 34 - 2017
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?

### Divisible Palindrome

##### KS 3 Short Challenge Level:

Weekly Problem 5 - 2014
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

### Times and Square

##### KS 3 Short Challenge Level:

Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?

##### KS 3 Short Challenge Level:

Weekly Problem 33 - 2015
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

### Jenny's Logic

##### KS 3 Short Challenge Level:

Weekly Problem 52 - 2009
How did Jenny figure out that Tom's cards added to an even number?

### Ones, Twos and Threes

##### KS 3 Short Challenge Level:

Weekly Problem 5 - 2017
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

##### KS 3 Short Challenge Level:

Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.

### Expanding Zeros

##### KS 3 Short Challenge Level:

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?

### Big Blackboard

##### KS 3 Short Challenge Level:

Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?

### Missing Digit

##### KS 3 Short Challenge Level:

Weekly Problem 37 - 2006
What digit must replace the star to make the number a multiple of 11?

### Square LCM

##### KS 3 Short Challenge Level:

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

### Colossal Sum

##### KS 3 Short Challenge Level:

What is the units digit in this sum of powers of 9?

### Powerful Finale

##### KS 3 Short Challenge Level:

Weekly Problem 24 - 2017
What is the last digit of $3^{2011}$?

### 17s and 23s

##### KS 3 Short Challenge Level:

Weekly Problem 22 - 2010
Can you form this 2010-digit number...

### Cinema Costs

##### KS 3 Short Challenge Level:

Weekly Problem 41 - 2009
At a cinema a child's ticket costs £4.20 and an adult's ticket costs £7.70. How much did is cost this group of adults and children to see a film?

### Cancelling Fractions

##### KS 3 Short Challenge Level:

Can you find a fraction with the following properties?

### Leap Monday

##### KS 3 Short Challenge Level:

Can you find the next time that the 29th of February will fall on a Monday?

### Four or Five

##### KS 3 Short Challenge Level:

Weekly Problem 19 - 2012
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

### Cakes and Buns

##### KS 3 Short Challenge Level:

Weekly Problem 41 - 2006
Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?

### Seven from Nine

##### KS 3 Short Challenge Level:

Weekly Problem 21 - 2013
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

### Supercomputer

##### KS 4 Short Challenge Level:

Weekly Problem 28 - 2014
What is the units digit of the given expression?

### Long List

##### KS 4 Short Challenge Level:

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?

### Triangular Algebra

##### KS 4 Short Challenge Level:

Weekly Problem 26 - 2017
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?

### HCF Expression

##### KS 4 Short Challenge Level:

Weekly Problem 30 - 2010
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

### Factor List

##### KS 4 Short Challenge Level:

Weekly Problem 48 - 2010
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

### Three Primes

##### KS 4 Short Challenge Level:

Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

### Factorised Factorial

##### KS 4 Short Challenge Level:

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

### Square Product

##### KS 4 Short Challenge Level:

Weekly Problem 10 - 2011
Will this product give a perfect square?

### Adding a Square to a Cube

##### KS 4 Short Challenge Level:

If you take a number and add its square to its cube, how often will you get a perfect square?

### Primes and Six

##### KS 4 Short Challenge Level:

Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.

### Integer and Integer

##### KS 4 Short Challenge Level:

Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?

### Fortunate Inflation

##### KS 4 Short Challenge Level:

The price of an item in pounds and pence is increased by 4%. The new price is an integer number of pounds. Can you find it?